Jinwu Ye

Penn State University

Outline of the talk:1. Introduction to Boson Hubbard Model and supersolids on lattices2. Boson -Vortex duality in boson Hubbard Model3. Extended Hubbard Model at honeycomb and square lattice near

half filling4. Charge Density Wave supersolid5. Valence Bond Supersolid6. Implications to QMC7. Conclusions

Quantum phase transitions from Solids to Supersolids in bi-partite lattices

1. The mobility of dual vortices in square, honeycomb, triangular, Kagome and Dice lattices. Longhua Jiang and Jinwu Ye, J. Phys, Condensed Matter. 18 (2006) 6907-6922

2. Quantum phase transitions from solids to supersolids in bipartite lattices Jinwu Ye, cond-mat/0503113. Revised version, submitted to Phys. Rev. Lett.

3. Supersolids and solids to Supersolids transitions in frustrated lattices , Jinwu Ye, cond-mat/0612009,

4. The Extended Bose Hubbard Model on the Two Dimensional Honeycomb Lattice Jing Yu Gan, Yu Chuan Wen, Jinwu Ye, Tao Li, Shi-Jie Yang, Yue Yu,

Phys. Rev. B 75, 214509 (2007)

References:

1. Introduction to extended boson Hubbard model and supersolid on a lattice

Superfluid to Mott insulator transition in optical lattice, Greiner et al, 2002

Ultra-cold alkali gases:

Extended Boson Hubbard Model (EBHM):

......

Ring exchange interaction:

t : nearest neighbor hopping

21,, VVU : onsite, nn and nnn interactions

Bose-Einstein condensation (BEC)

Off-diagonal long range order: 0 ib

Diagonal order in boson density: iii bbn

(1) Superfluid:

(2) Charge density wave:

(3) Supersolid:

Both off-diagonal and diagonal long range order

Rather complete quantum phenomenology was worked out in

Jinwu Ye, Phys. Rev. Lett. 97, 125302 (2006).

Jinwu Ye, cond-mat/0705.0770, submitted to Nature Physics

Jinwu Ye, cond-mat/0603269, submitted to Phys. Rev. B.

Kim and Chan, 2005

Supersolids on lattices are very different than possible supersolids in Helium 4

See my seminar at KITP-C on Aug. 8, 2007

At Integer fillings , Superconductor to Mott insulator transition ( SIT ) was studied in:

• Onsite interaction only: Fisher, Weichman, Grinstein and Fisher, 1989• Long-range Coulomb interaction: Fisher and Grinstein, 1988; Jinwu Ye, 1998

2. Boson-Vortex duality in EBHM at filling factor q

pf

nf

Boson-Vortex duality at 1f

Particles ~

Holes ~

Number of particles= number of holes relativistic field theory for

]|||||||[| 42222 usrddS r Insulator: < > =0; Superfluid < > 0

Excitations of the superfluid:(1) Gapless Phonons: represented by a dual U(1) gauge field (2) Gapful Vortices: vortex operator

Number of vortices = number of antivortices relativistic field theory for

A

])(4

1|||||)([| 24222

AusiArddS ddvor

Superfluid: < > =0; Insulator: < > 0

Superfluid:< > 0< > = 0

]|||||||[| 42222 usrddS r Boson quasi-particle excitations ~ in the insulator :

is dual to

Vortex excitations ~ + phonon excitations in the superfluid:

])(4

1|||||)([| 24222

AusiArddS ddvor

A

Insulator: < > =0 < > 0

Boson-Vortex duality and SIT at 1f

Dasgupta and Halperin, 1981

SIT at cold alkali gas: Greiner et al, 2002

A vortex in the vortex field is the original boson, the vortex acquires a phase when it encircles a boson

The average strength of the dual magnetic field on the vortex field = density of boson= flux quanta per plaquette

2

f

The boson acquires a phase when it encircles a vortex

2

MSG dictates that there are at least q -fold degenerate minima which forms a q dimensional representation of the MSG.

The effective theory in terms of these q vortices should be invariant under this MSG.

Bosons at filling factor f=p/q hopping on a direct lattice

is dual to

Vortices hopping on the dual lattice subject to a fluctuating dual " magnetic field“ whose average strength is f=p/q flux quanta per dual plaquette.

This is similar to

Hofstadter problem of electrons moving in a lattice in the presence of f=p/q flux quanta per plaquette. Hofstadter , 1976

The spectrum can be classified by Magnetic Space Group ( MSG)

Balents, Bartosch, Burkov, Sachdev, Sengupta, 2005

Balents, Bartosch, Burkov, Sachdev, Sengupta, 2005

Taking and apply to study CDW state in high temperature superconductors

,........32,16q

Two holes in the DVM developed by Balents et.al 2005

(1) It can NOT be applied to the CDW side:

In the CDW side, the boson density takes, for example, a checkboard ( or stripe ) Order.

The average strength of the dual magnetic field per dual plaquetteshould also be taken the same pattern ! Not as uniform as

The average strength of the dual magnetic field on the vortex field = density of boson

qpf /

,........32,16q

(2) As shown in: Longhua Jiang and Jinwu Ye, 2006

The bandwidth of the dual vortex band cqAew

The DVM completely breaks down when

1. I take a completely different route:

Taking , so the DVM works, then moving slightly awayfrom , the return is interesting and important:

Supersolids could exist only at in-commensurate fillings !

2q2q

2. We will also extend the DVM also to the CDW side

3. EBHM on honeycomb lattice near half filling

Bosons hopping on honeycomb lattice at filling factor f=p/q

Vortices hopping on triangular lattice subject to a fluctuating magnetic field whose average strength is f=p/q per triangular plaquette

is dual to

The Hamiltonian of the vortices is:

vt is the vortex hopping amplitude

+ interaction terms

The space group of the triangular lattice:(1) Point group: , 12 elements with 2 generators

(2) Two translations

66 ~ DC v 13/ , IR

21,TT

The Magnetic Space Group (MSG ) are generated by:

Translations and Rotations: Reflections:

fTTTTTTT d 2;, 122

2121 is the complex congugateK

They all commute with the Hamiltonian H

The rotations around the two direct lattice points A and B:

In the following, we focus on q=2 where the energy band is:

There are two minima at where the two eigenmodes are labeled by

Both also commute with H

transform under the MSG as:

The quadratic terms of the effective action is the scalar electrodynamics:

The most general quartic term invariant under all the transformations is:

Moving slightly away from half filling f=1/2 corresponds to adding a small dual magnetic field in the action:2

1 ff

r > 0, superfluid state,

< > =0, l l

r < 0, insulating state, < > for at least one l1. : Ising limit2. : Easy-plane limit

l 001 01

In the following, we focus on the insulating side.

It turns out that the action only works in the SF and VBS sides,But breaks down in the CDW side

The SF is stable against the changing of chemical potential

In the Ising limit, at mean field:

Density Wave order parameter in the Ising limit:

Saddle point for the dual gauge field:

for sublattice A

for sublattice B

A and B could be checkboard or stripe order

4. Charge density wave supersolid

A uniform saddle point does not work in the CDW side

There is only one vortex minimum in such a staggered dual magnetic field with 2/1

Slightly away from 1/2 filling, the effective action inside the CDW is:

It has a structure identical to q=1 component Ginzburg-Landau model for a type-II superconductor in a magnetic field

aA

is always massive, so does not appear in above Eqn.

Phase diagram of Type II superconductorin external magnetic field:

Fisher, Fisher and Huse, 1991

D. Nelson, 1989

Ising limit

C-CDW ]2

1)[11(

nf

Fisher, Weichman, Grinstein and Fisher, 1989

IC-CDW

The CDW could also be stripe order

ff 2/1

In the CDW state: bA

is massive

In the CDW-SS state:

0 b

0 b

bA

becomes gapless which stands for the superfluid mode

The transition from the CDW to the CDW-SS is in the same universality class as that from the SF to Mott insulator,Therefore have exact critical exponents:

0,2/1,2 z

The superfluid density in the CDW-SS scales as:

with a logarithmic correction

It is important to compare with microscopic calculations suchas quantum monte-carlo (QMC)

Similar phase diagram also holds for square lattice

The IC-CDW can be stabilized onlyby very long-range interactions

The transition from the SF to the CDW-SS is first order

The transition from the CDW-SSto the IC-CDW is first order

The CDW-SS has the same diagonal density order as the CDW at 2/1f

QMC simulations by:

F. Hebert, et.al, Phys. Rev. B, 65, 014513 (2001).

The stripe solid to stripe-SS transition is 2nd order with

in the stripe supersolid, but scale in the same waywith different coefficients

0, 1 VU 0,0, 21 VVU

Soft core case:

P. Sengupta, et.al. 2005

),( SS is stable

Hard core

In the easy-plane limit:

Valence Bond Solid (VBS) order parameter:

latticedualxQxQK AB ,),1,1(3

2),cos(

At mean field:

The uniform saddle point holds in bothSF and the VBS

5.Valence bond supersolid

Sixth order terms are neededto determine the relative phase

Only can fix the relative phase It turns out both signs of are equivalent

,0 The other two VBS can be obtained by

BAR /3/2

P.W. Anderson; Science 235, 1196(1987)

A ring exchange term :

is needed to stabilize the VBS order

is relevant, so the SF to VBS transition is 1st order 3cos3 C

3cos3 C

Slightly away from half-filling inside the VBS:

sector is massive, so can be integrated out

Assuming ,0

It is also identical to q=1 component GL model for a type-II superconductor in a magnetic field

02~ u

All the discussions on the Ising case follow

In the VBS:

,0 ba

In the VB-SS:

0 ba

Gapless superfluid mode: A

A is massive

The lowest order term coupling the two phases is

)0(0 VBS has Columnar dimer ( plaquette ) pattern

sqC is irrelevant, so the SF to the VBS is a second order transition through so called deconfined QCPdiscovered by O. Motrunich and A. Vishwanath, 2004

T. Senthil et. al, 2004

There are strong numerical evidence that there is no such deconfined QCP, the SF to the VBS is weakly 1st order

Anatoly Kuklov, Nikolay Prokof'ev, Boris Svistunov, M. Troyer, con-mat/0602466

In square lattice:

(1) In square lattice: Stripe solid to stripe supersolid in the hard core case, Checkboard solid to checkboard supersolid in soft case to test the universality class by finite size scaling

0,2/1,2 z

(2) Do the same things in the honeycome lattice

(3) With ring-exchange term, test the VBS to VB-SS transition

Static structure factor:

Equal time

K: tuning parameter:

Similar quantities can be defined for a bond ..chbbK jiij

:Q

is the ordering wave vector

6. Implications on future QMC

Jing Yu Gan, Yu Chuan Wen, Jinwu Ye, Tao Li, Shi-Jie Yang, Yue Yu, Phys. Rev. B 75, 214509 (2007)

Soft core boson in honeycomb lattice

7. Conclusions

These results should have direct impacts on ongoing experiments of cold atoms loaded on optical lattices. It should also shed some lights on possible supersolid Helium 4.

(1) Map out the global phase diagram in a unified theory: SF, CDW( VBS), CDW-SS (VB-SS), IC-CDW (IC-VBS)

(2) Discover universality class of solids to supersolids transitions

(3) Propose a novel kind of supersolid: valence bond supersolid

(4) Several important implications on QMC